Carlos and Ana were asked to find an explicit formula for the sequence $-10,-4,2,8,...$, where the first term should be $f(1)$. Carlos said the formula is $f(n)=-10+6n$. Ana said the formula is $f(n)=-10+6(n-1)$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Carlos (Choice B) B Only Ana (Choice C) C Both Carlos and Ana (Choice D) D Neither Carlos nor Ana
The general explicit formula for arithmetic sequences is ${a_1}+{d}(n-1)$, where ${a_1}$ is the first term and $ d$ is the common difference. The first term is ${-10}$ and the common difference is ${6}$. ${+6\,\curvearrowright}$ ${+6\,\curvearrowright}$ ${+6\,\curvearrowright}$ ${-10},$ $-4,$ $2,$ $8,...$ We get the following formula. $f(n)={-10}+{6}(n-1)$ So Ana is definitely right. What about Carlos? We can see that $f(n)=-10+6n$ is not a correct formula, because the constant difference is added one extra time for each term. For instance, according to this formula, the value of the first term would be: $f(1)=-10+6\cdot1=-4$. However, according to our table of values, $f(1)=-10$. So Carlos is definitely wrong. Only Ana got a correct explicit formula.